Solve for $x$ in the equation $x^2-12 x+36=90$.
A. $x=6 \pm 3 \sqrt{10}$
B. $x=6+2-\sqrt{7}$
C. $x=12+3 \sqrt{22}$
D. $x=12 \pm 3 \sqrt{10}$
Answer (2)
Recognize the perfect square trinomial: Rewrite the equation as ( x − 6 ) 2 = 90 .
Take the square root of both sides: Obtain x − 6 = ± 90 .
Simplify the square root: Simplify 90 to 3 10 .
Solve for x : Find the solutions x = 6 ± 3 10 .
Explanation
Analyze the equation We are given the equation x 2 − 12 x + 36 = 90 . Our goal is to solve for x . Notice that the left side of the equation is a perfect square trinomial.
Rewrite the equation We can rewrite the left side as ( x − 6 ) 2 , so the equation becomes ( x − 6 ) 2 = 90 .
Take the square root Now, we take the square root of both sides of the equation: ( x − 6 ) 2 = ± 90 . This gives us x − 6 = ± 90 .
Simplify the square root We can simplify 90 as 9 ⋅ 10 = 9 ⋅ 10 = 3 10 . So, we have x − 6 = ± 3 10 .
Solve for x Finally, we solve for x by adding 6 to both sides: x = 6 ± 3 10 .
State the solutions Therefore, the solutions for x are x = 6 + 3 10 and x = 6 − 3 10 .
Examples
Imagine you are designing a square garden and want to increase its area by a certain amount. This problem is similar to finding the new side length of the garden after increasing its area. Understanding how to solve quadratic equations like this can help you determine the dimensions needed for your garden or any other square-shaped area you are working with. This type of problem also appears in physics, when calculating distances or velocities.
The solutions to the equation x 2 − 12 x + 36 = 90 are found by recognizing it as a perfect square, then taking the square root and simplifying. This leads to the final result x = 6 ± 3 10 , which corresponds to option A. Therefore, the answer is A: x = 6 ± 3 10 .
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